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Issues with understanding P-values

After doing a couple of questions I realised that I have not fully grasped the concept of understanding the rejection region step of small-sample inferences aswell as Large-sample tests.

the **P-value** method is quite confusing, perticularly when the two-sided tailed tests are put into play. My knowledge of critical value is also effected by it. I know about alpha and the significance level used for the test. Its basically the comparing the test calculated values with the alpha for determining the rejection region, that is confusing me.

Here are some questions that are using the P-values in different ways which is the primary root of my problem

EDIT:

I can't post the questions since they are lengthy and I don't have a scanner at the moment but here is the solutions which show how the p-value is taken.

solution manuel:

Attachment 17951:

If it comes to requiring the questions, I would post them sooner or later.